(* Basic forward lemmas *****************************************************)
-lemma ylt_inv_gen: ∀x,y. x < y → ∃m. x = yinj m.
+lemma ylt_fwd_gen: ∀x,y. x < y → ∃m. x = yinj m.
#x #y * -x -y /2 width=2 by ex_intro/
qed-.
+lemma ylt_fwd_lt_O1: ∀x,y:ynat. x < y → 0 < y.
+#x #y #H elim H -x -y /3 width=2 by ylt_inj, ltn_to_ltO/
+qed-.
+
(* Basic inversion lemmas ***************************************************)
fact ylt_inv_inj2_aux: ∀x,y. x < y → ∀n. y = yinj n →
qed-.
lemma ylt_inv_Y1: ∀n. ∞ < n → ⊥.
-#n #H elim (ylt_inv_gen … H) -H
+#n #H elim (ylt_fwd_gen … H) -H
#y #H destruct
qed-.
(* inversion and forward lemmas on yle **************************************)
-lemma lt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
-#m #n * -m -n /3 width=1 by yle_pred_sn, yle_inj, yle_Y/
+lemma ylt_fwd_le_succ1: ∀m,n. m < n → ⫯m ≤ n.
+#m #n * -m -n /2 width=1 by yle_inj/
+qed-.
+
+lemma ylt_fwd_le_pred2: ∀x,y:ynat. x < y → x ≤ ⫰y.
+#x #y #H elim H -x -y /3 width=1 by yle_inj, monotonic_pred/
+qed-.
+
+lemma ylt_fwd_le: ∀m:ynat. ∀n:ynat. m < n → m ≤ n.
+#m #n * -m -n /3 width=1 by lt_to_le, yle_inj/
qed-.
lemma ylt_yle_false: ∀m:ynat. ∀n:ynat. m < n → n ≤ m → ⊥.
#H destruct
qed.
+(* Properties on predecessor ************************************************)
+
+lemma ylt_pred: ∀m,n. m < n → 0 < m → ⫰m < ⫰n.
+#m #n * -m -n
+/4 width=1 by ylt_inv_inj, ylt_inj, monotonic_lt_pred/
+qed.
+
(* Properties on successor **************************************************)
lemma ylt_O_succ: ∀n. 0 < ⫯n.
qed.
lemma ylt_succ: ∀m,n. m < n → ⫯m < ⫯n.
-#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
+#m #n #H elim H -m -n /3 width=1 by ylt_inj, le_S_S/
+qed.
+
+lemma yle_succ1_inj: ∀x,y. ⫯yinj x ≤ y → x < y.
+#x * /3 width=1 by yle_inv_inj, ylt_inj/
qed.
(* Properties on order ******************************************************)
-lemma yle_split_eq: ∀m:ynat. ∀n:ynat. m ≤ n → m < n ∨ m = n.
+lemma yle_split_eq: ∀m,n:ynat. m ≤ n → m < n ∨ m = n.
#m #n * -m -n
[ #m #n #Hmn elim (le_to_or_lt_eq … Hmn) -Hmn
/3 width=1 by or_introl, ylt_inj/
]
qed-.
-lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m..
+lemma ylt_split: ∀m,n:ynat. m < n ∨ n ≤ m.
#m #n elim (yle_split m n) /2 width=1 by or_intror/
#H elim (yle_split_eq … H) -H /2 width=1 by or_introl, or_intror/
-qed-.
+qed-.
+
+lemma ylt_split_eq: ∀m,n:ynat. ∨∨ m < n | n = m | n < m.
+#m #n elim (ylt_split m n) /2 width=1 by or3_intro0/
+#H elim (yle_split_eq … H) -H /2 width=1 by or3_intro1, or3_intro2/
+qed-.
lemma ylt_yle_trans: ∀x:ynat. ∀y:ynat. ∀z:ynat. y ≤ z → x < y → x < z.
#x #y #z * -y -z